Algebraic genericity and the differentiability of the convolution
The convolution of the $2$-periodic functions $f, g \in L^1[-1, 1]$ is defined as the function $(f * g)(x) = \int_{-1}^1 f(s)g(x-s)ds$. It is well known that, for instance, if $f \in C^k[-1,1]$ then $(f*g) \in C^k[-1,1]$ without any kind of requirement on $g$. However, there are examples of two differentiable functions whose convolution is not differentiable. \par In this interesting paper, the authors focus their efforts on lineability, algebrability, and coneability within this context, that is, they look for the existence of large algebras of the biggest possible dimension formed by differentiable functions (with the null vector not included) whose convolution is not differentiable. More specifically, they prove that there exist two algebras $V$ and $W$ (both of dimension $\mathfrak{c}$) in $D[-1, 1]$ ($2$-periodic differentiable functions) so that, if $f \in V \setminus \{0\}$ and $g \in W \setminus \{0\}$, then $(f * g)$ is not differentiable at $0$. \par In particular, they show that every function in $C[-1, 1]$ (continuous $2$-periodic functions) can be approximated by vectors taken from a linear space consisting of differentiable functions that (except for 0) may be paired with another differentiable function so that the resulting convolution is not differentiable at $0$.
scientific article; zbMATH DE number 7043341
scientific article